Many open question do remain. First, it would be interesting to generalize Theorems REF  and REF  to other feature distributions than the isotropic Gaussian measure, for instance heavy-tailed features as in [1]}, sub-Gaussian features [2]} or correlated Gaussian features [3]}. The main difficulty to overcome is an extension of Lemma REF  to these cases. Moreover, while the convergence rate in Corollary REF  is among the best available results if \(\beta ^*\)  is allowed to be genuinely effectively sparse, it is not clear whether the exponent in (REF ) is optimal, and further research about information theoretic lower bounds is needed. When \(\beta ^*\)  is exactly sparse, the convergence rate in (REF ) is sub-optimal and better results in terms of the fraction of mislabeled data points and the sparsity have been obtained in [4]}. In particular, for noiseless data and exact sparse \(\beta ^*\)  our simulations suggest that AdaBoost attains a faster rate with exponent \(1/2\)  instead of \(1/3\)  whereas in the noisy case the exponent \(1/3\)  seems to be optimal. It is an interesting further research question how to show that AdaBoost attains faster convergence rates for noiseless data and when \(\beta ^*\)  is exact sparse.
